A037143 -id:A037143 - cp's OEIS frontend

A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 0, 0, 1, 1, 0, 1, 0, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 1, 0, 1, 1, 0, 0, 3, 0, 1, 0, 1, 0, 1, 0, 2, 0, 0, 0, 4, 0, 0, 1, 2, 0, 1, 0, 1, 0, 1, 0, 4, 0, 0, 1, 1, 0, 1, 0, 3, 1, 0, 0, 4, 0, 0, 0, 2, 0, 2, 0, 1, 0, 0, 0, 4, 0, 1, 1, 2, 0, 1, 0, 2, 1, 0, 0, 4, 0, 1, 0, 3, 0, 1, 0, 1, 1, 0, 0, 5

Offset: 1

Gus Wiseman, Oct 14 2021

nonn

This is also the number of distinct possible alternating products of length-3 factorizations of n, where we define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)), and where a factorization of n is a weakly increasing sequence of positive integers > 1 with product n.

			Representative factorizations for each of the a(360) = 9 alternating products:
   (2,2,90) -> 90
   (2,3,60) -> 40
   (2,4,45) -> 45/2
   (2,5,36) -> 72/5
   (2,6,30) -> 10
   (2,9,20) -> 40/9
  (2,10,18) -> 18/5
  (2,12,15) -> 5/2
   (3,8,15) -> 45/8

Antti Karttunen, Table of n, a(n) for n = 1..65537

Allowing factorizations of any length <= 3 gives A033273.
Positions of positive terms are A033942.
Positions of 0's are A037143.
The length-2 version is A072670.
Allowing any length gives A347460, reverse A038548.
Allowing any odd length gives A347708.
A001055 counts factorizations (strict A045778, ordered A074206).
A122179 counts length-3 factorizations.
A292886 counts knapsack factorizations, by sum A293627.
A301957 counts distinct subset-products of prime indices.
A304792 counts distinct subset-sums of partitions, positive A276024.
Cf. A000040, A001358, A002033, A046951, A339846, A339890, A347437, A347438, A347439, A347440, A347442, A347456, A347461.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
Table[Length[Union[altprod/@Select[facs[n],Length[#]==3&]]],{n,100}]

A347709(n) = { my(rats=List([])); fordiv(n,z,my(yx=n/z); fordiv(yx, y, my(x = yx/y); if((y <= z) && (x <= y) && (x > 1), listput(rats,x*z/y)))); #Set(rats); }; \\ Antti Karttunen, Jan 29 2025

More terms from Antti Karttunen, Jan 29 2025

A368728 Numbers whose prime indices are 1, prime, or semiprime.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 58, 59, 60, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 73, 75

Offset: 1

Gus Wiseman, Jan 07 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

These are products of primes indexed by elements of A037143.
For just primes we have A076610, strict A302590.
For just semiprimes we have A339112, strict A340020.
For squarefree semiprimes we have A339113, strict A309356.
The odd case is A368729, strict A340019.
The complement is A368833.
A000607 counts partitions into primes, A034891 with ones allowed.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628, A320911, A320912.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Max@@Length/@prix/@prix[#]<=2&]

Closed under multiplication.

A368729 Numbers whose prime indices are prime or semiprime. MM-numbers of labeled multigraphs with loops and half-loops without isolated (uncovered) nodes.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 15, 17, 21, 23, 25, 27, 29, 31, 33, 35, 39, 41, 43, 45, 47, 49, 51, 55, 59, 63, 65, 67, 69, 73, 75, 77, 79, 81, 83, 85, 87, 91, 93, 97, 99, 101, 105, 109, 115, 117, 119, 121, 123, 125, 127, 129, 135, 137, 139, 141, 143, 145, 147, 149

Offset: 1

Gus Wiseman, Jan 07 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.

			The terms together with the corresponding multigraphs begin:
   1: {}
   3: {{1}}
   5: {{2}}
   7: {{1,1}}
   9: {{1},{1}}
  11: {{3}}
  13: {{1,2}}
  15: {{1},{2}}
  17: {{4}}
  21: {{1},{1,1}}
  23: {{2,2}}
  25: {{2},{2}}
  27: {{1},{1},{1}}
  29: {{1,3}}
  31: {{5}}
  33: {{1},{3}}
  35: {{2},{1,1}}
  39: {{1},{1,2}}
  41: {{6}}
  43: {{1,4}}
  45: {{1},{1},{2}}
  47: {{2,3}}
  49: {{1,1},{1,1}}

In the unlabeled case these multigraphs are counted by A320663.
These are products of primes indexed by elements of A037143 greater than 1.
For just primes we have A076610, squarefree A302590.
For just semiprimes we have A339112, squarefree A340020.
For just half-loops we have A340019.
This is the odd case of A368728, complement A368833.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320912, A339113.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],OddQ[#]&&Max@@Length/@prix/@prix[#]<=2&]

A368732 Primes whose index is one, another prime number, or a semiprime.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 23, 29, 31, 41, 43, 47, 59, 67, 73, 79, 83, 97, 101, 109, 127, 137, 139, 149, 157, 163, 167, 179, 191, 199, 211, 227, 233, 241, 257, 269, 271, 277, 283, 293, 313, 331, 347, 353, 367, 373, 389, 401, 421, 431, 439, 443, 449, 461, 467, 487

Offset: 1

Gus Wiseman, Jan 08 2024

nonn

For just primes we have A006450, products A076610, strict A302590.
These indices are A037143.
For just semiprimes we have A106349, products A339112, strict A340020.
Products of these primes are A368728, odd A368729, odd strict A340019.
Products of the complementary primes are A368833.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
A322551 lists primes of squarefree semiprime index.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A309356, A320628, A320911, A320912, A339113.

Prime/@Select[Range[100],PrimeOmega[#]<=2&]

A368833 Numbers whose prime indices are not 1, prime, or semiprime.

Original entry on oeis.org

19, 37, 38, 53, 57, 61, 71, 74, 76, 89, 95, 103, 106, 107, 111, 113, 114, 122, 131, 133, 142, 148, 151, 152, 159, 171, 173, 178, 181, 183, 185, 190, 193, 197, 206, 209, 212, 213, 214, 222, 223, 226, 228, 229, 239, 244, 247, 251, 259, 262, 263, 265, 266, 267

Offset: 1

Gus Wiseman, Jan 08 2024

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

			The terms together with their prime indices begin:
   19: {8}
   37: {12}
   38: {1,8}
   53: {16}
   57: {2,8}
   61: {18}
   71: {20}
   74: {1,12}
   76: {1,1,8}
   89: {24}
   95: {3,8}
  103: {27}
  106: {1,16}
  107: {28}
  111: {2,12}
  113: {30}
  114: {1,2,8}
  122: {1,18}
  131: {32}
  133: {4,8}
  142: {1,20}
  148: {1,1,12}

These are non-products of primes indexed by elements of A037143.
The complement for just primes is A076610, strict A302590.
The complement for just semiprimes is A339112, strict A340020.
The complement for just squarefree semiprimes is A339113, strict A309356.
The complement is A368728.
The complement for just primes and semiprimes is A368729, strict A340019.
A000607 counts partitions into primes, with ones allowed A034891.
A001358 lists semiprimes, squarefree A006881.
A006450, A106349, A322551, A368732 list selected primes.
A056239 adds up prime indices, row sums of A112798.
A101048 counts partitions into semiprimes.
Cf. A000040, A000720, A001222, A003963, A005117, A302242, A320628.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100], Max@@PrimeOmega/@prix[#]>2&]

A121615 Lexicographically earliest sequence such that the k-th prime occurs altogether not more than k times as prime factor.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 15, 17, 19, 23, 29, 31, 35, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 211, 221, 223, 227, 229, 233, 239

Offset: 1

Reinhard Zumkeller, Aug 11 2006

nonn

Subsequence of A037143: 1 <= A001222(a(n)) <= 2; A000040 is a subsequence.

			k=1, prime(1)=2: {2} = {a(1)},
k=2, prime(2)=3: {3,3*5} = {a(2),a(7)},
k=3, prime(3)=5: {5,3*5,5*7} = {a(3),a(7),a(13)},
k=4, prime(4)=7: {7,5*7,7*7} = {a(4),a(7),a(18)},
k=5, prime(5)=11: {11,11*11,11*13,11*17} = {a(5),a(34),a(39),a(49)},
k=6, prime(6)=13: {13,11*13,13*13,13*17,13*19} = {a(6),a(39),a(45),a(55),a(62)};
k=7, prime(7)=17: {17,11*17,13*17,17*17,17*19,17*23} = {a(8),a(49),a(55),a(71),a(77),a(90)}.

R. Zumkeller, Table of n, a(n) for n = 1..1000

A166719 Numbers with at most 5 prime factors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73

Offset: 1

Michael B. Porter, Oct 20 2009

Complement of A046305, A001222(a(n))<=5

			50 = 2*5*5 is in the sequence since it has 3 prime factors and 3 <= 5
64 = 2*2*2*2*2*2 is not in the sequence since it has 6 prime factors

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A046305, A001222

For numbers with at most n prime factors: n=1: A000040, n=2: A037143, n=3: A037144, n=4: A166718.

Select[Range[100],PrimeOmega[#]<6&] (* Harvey P. Dale, Jul 13 2011 *)

PARI
```
isA166719(n) = (bigomega(n) <= 5)
```

A176540 1 together with the semiprimes.

Original entry on oeis.org

1, 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187

Offset: 1

Juri-Stepan Gerasimov, Apr 19 2010

nonn

The products of two noncomposite numbers that do not contain prime numbers.

Cf. A001358, A037143.

Join[{1},Select[Range[200],PrimeOmega[#]==2&]] (* Harvey P. Dale, Dec 31 2015 *)

Definition and comment swapped by Michel Marcus, Aug 09 2014

A276830 Number of ways to write n as ((p-1)/2)^2 + P_2, where p is an odd prime and P_2 is a product of at most two primes.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 2, 2, 1, 3, 3, 2, 2, 3, 3, 2, 1, 3, 2, 2, 1, 2, 3, 2, 1, 4, 3, 2, 2, 4, 2, 3, 1, 3, 4, 2, 2, 5, 4, 4, 2, 5, 3, 3, 2, 3, 5, 3, 1, 5, 3, 2, 2, 2, 3, 3, 2, 4, 4, 3, 2, 5, 3, 2, 3, 5, 3, 4, 3, 4, 5, 2, 3, 5, 4, 2, 3, 5, 2, 3

Offset: 1

Zhi-Wei Sun, Sep 20 2016

nonn

Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 9, 17, 21, 25, 33, 49, 109, 169, 189, 361, 841, 961, 12769, 19321.

See also A276825 for a similar conjecture involving cubes, and some comments on x^2 + P_2.

			a(2) = 1 since 2 = ((3-1)/2)^2 + 1 with 3 prime.
a(3) = 1 since 3 = ((3-1)/2)^2 + 2 with 3 and 2 both prime.
a(4) = 1 since 4 = ((3-1)/2)^2 + 3 with 3 prime.
a(9) = 1 since 9 = ((5-1)/2)^2 + 5 with 5 prime.
a(17) = 1 since 17 = ((5-1)/2)^2 + 13 with 5 and 13 both prime.
a(21) = 1 since 21 = ((5-1)/2)^2 + 17 with 5 and 17 both prime.
a(25) = 1 since 25 = ((5-1)/2)^2 + 3*7 with 5, 3 and 7 all prime.
a(33) = 1 since 33 = ((5-1)/2)^2 + 29 with 5 and 29 both prime.
a(49) = 1 since 49 = ((13-1)/2)^2 + 13 with 13 prime.
a(109) = 1 since 109 = ((13-1)/2)^2 + 73 with 13 and 73 both prime.
a(169) = 1 since 169 = ((13-1)/2)^2 + 7*19 with 13, 7 and 19 all prime.
a(189) = 1 since 189 = ((5-1)/2)^2 + 5*37 with 5 and 37 both prime.
a(361) = 1 since 361 = ((37-1)/2)^2 + + 37 with 37 prime.
a(841) = 1 since 841 = ((37-1)/2)^2 + 11*47 with 37, 11 and 47 all prime.
a(961) = 1 since 961 = ((61-1)/2)^2 + 61 with 61 prime.
a(12769) = 1 since 12769 = ((109-1)/2)^2 + 59*167 with 109, 59 and 167 all prime.
a(19321) = 1 since 19321 = ((277-1)/2)^2 + 277 with 277 prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000040, A000290, A037143, A235645, A276711, A276825.

PP[n_]:=PP[n]=PrimeQ[Sqrt[n]]||(SquareFreeQ[n]&&Length[FactorInteger[n]]<=2)
Do[r=0;Do[If[PP[n-((Prime[k]-1)/2)^2],r=r+1;If[r>1,Goto[aa]]],{k,2,PrimePi[2*Sqrt[n]+1]}];Print[n," ",r];
Label[aa];If[Mod[n,50000]==0,Print[n]];Continue,{n,10^5,1000000}]

A278351 Least number that is the start of a prime-semiprime gap of size n.

Original entry on oeis.org

2, 7, 26, 97, 341, 241, 6091, 3173, 2869, 2521, 16022, 26603, 114358, 41779, 74491, 39343, 463161, 104659, 248407, 517421, 923722, 506509, 1930823, 584213, 2560177, 4036967, 4570411, 4552363, 7879253, 4417813, 27841051, 5167587, 13683034, 9725107, 47735342, 25045771, 63305661

Offset: 1

Bobby Jacobs, Charles R Greathouse IV, Jonathan Vos Post, and Robert G. Wilson v, Nov 23 2016

nonn

A prime-semiprime gap of n is defined as the difference between p & q, p being either a prime, A000040, or a semiprime, A001358, and q being the next greater prime or semiprime, see examples.
The corresponding numbers at the end of the prime-semiprime gaps, i.e., a(n)+n, are in A278404.
In the first 52 terms, 19 are primes and the remaining 33 are semiprime. Of the end-of-gap terms a(n)+n, 20 are primes and 32 are not. There are only 6 pairs of p and q that are both primes, and 19 pairs that are both semiprime.

			a(1) = 2 since there is a gap of 1 between 2 and 3, both of which are primes.
a(2) = 7 since there is a gap of 2 between 7 and 9, the first is a prime and the second is a semiprime.
a(3) = 26 since there is a gap of 3 between 26, a semiprime, and 29, a prime.
a(6) = 241 because the first prime-semiprime gap of size 6 is between 241 and 247.

Dana Jacobsen, Table of n, a(n) for n = 1..106 (first 52 terms from Bobby Jacobs, Charles R Greathouse IV, Jonathan Vos Post and Robert G. Wilson v)

Cf. A000230, A037143, A131109, A275108, A275013, A275014, A278404.

nxtp[n_] := Block[{m = n + 1}, While[ PrimeOmega[m] > 2, m++]; m]; gp[_] = 0; p = 2; While[p < 1000000000, q = nxtp[p]; If[ gp[q - p] == 0, gp[q -p] = p; Print[{q -p, p}]]; p = q]; Array[gp, 40]

use ntheory ":all";
my($final,$p,$nextn,@gp) = (40,2,1);  # first 40 values in order
forfactored {
  if (scalar(@) <= 2) { my $q = $;
    if (!defined $gp[$q-$p]) {
      $gp[$q-$p] = $p;
      while ($nextn <= $final && defined $gp[$nextn]) {
        print "$nextn $gp[$nextn]\n";
        $nextn++;
      }
      lastfor if $nextn > $final;
    }
    $p = $q;
  }
} 3,10**14; # Dana Jacobsen, Sep 10 2018

cp's OEIS Frontend

A347709 Number of distinct rational numbers of the form x * z / y for some factorization x * y * z = n, 1 < x <= y <= z.

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A368728 Numbers whose prime indices are 1, prime, or semiprime.

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A368729 Numbers whose prime indices are prime or semiprime. MM-numbers of labeled multigraphs with loops and half-loops without isolated (uncovered) nodes.

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A368732 Primes whose index is one, another prime number, or a semiprime.

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A368833 Numbers whose prime indices are not 1, prime, or semiprime.

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A121615 Lexicographically earliest sequence such that the k-th prime occurs altogether not more than k times as prime factor.

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A166719 Numbers with at most 5 prime factors (counted with multiplicity).

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PARI

A176540 1 together with the semiprimes.

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Extensions

A276830 Number of ways to write n as ((p-1)/2)^2 + P_2, where p is an odd prime and P_2 is a product of at most two primes.

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